PSLE Learner’s Guide · Volume 0858
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For every whole number n, n³ − n factors into three consecutive integers and is divisible by 6.
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This guide uses original Primary 6 practice and does not predict a live question. Consult the SEAB PSLE formats examined in 2026 page for official information.
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CHAPTER 1 OF 12 · Back to contents
1. Test the pattern
For n=2, 2³−2=6; for n=3, 27−3=24; for n=4, 64−4=60. Each result is divisible by 6, but a general proof must explain every whole number.
Build the explanation by naming the difference n cubed minus n and the factorisation n times n minus one times n plus one. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.
Repair the nearest misconception by checking three consecutive integers and returning to one factor 2 and one factor 3. State why the tempting alternative is too strong, incomplete or about a different stage.
- Check the difference n cubed minus n.
- Check the factorisation n times n minus one times n plus one.
- Check three consecutive integers.
- Check one factor 2 and one factor 3.
CHAPTER 2 OF 12 · Back to contents
2. Factor the expression
Use the difference of squares: n³−n=n(n²−1)=n(n−1)(n+1). Reorder the factors as (n−1)n(n+1) to reveal three consecutive integers.
Build the explanation by naming the difference n cubed minus n and the factorisation n times n minus one times n plus one. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.
Repair the nearest misconception by checking three consecutive integers and returning to one factor 2 and one factor 3. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 3 OF 12 · Back to contents
3. Find the factor 2
Among any three consecutive whole numbers, at least one is even. Their product therefore contains a factor 2.
Build the explanation by naming the difference n cubed minus n and the factorisation n times n minus one times n plus one. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.
Repair the nearest misconception by checking three consecutive integers and returning to one factor 2 and one factor 3. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 4 OF 12 · Back to contents
4. Find the factor 3
Among any three consecutive whole numbers, exactly one is divisible by 3. Their product therefore contains a factor 3.
Build the explanation by naming the difference n cubed minus n and the factorisation n times n minus one times n plus one. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.
Repair the nearest misconception by checking three consecutive integers and returning to one factor 2 and one factor 3. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 5 OF 12 · Back to contents
5. Combine coprime factors
Because the product contains both a factor 2 and a factor 3, it contains their product 6. Thus n³−n is divisible by 6.
Build the explanation by naming the difference n cubed minus n and the factorisation n times n minus one times n plus one. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.
Repair the nearest misconception by checking three consecutive integers and returning to one factor 2 and one factor 3. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 6 OF 12 · Back to contents
6. Keep zero and small cases valid
At n=0 or n=1, the expression equals zero. Zero is divisible by 6 because 0=6×0, so the rule still holds.
Build the explanation by naming the difference n cubed minus n and the factorisation n times n minus one times n plus one. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.
Repair the nearest misconception by checking three consecutive integers and returning to one factor 2 and one factor 3. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 7 OF 12 · Back to contents
7. Use the identity for remainders
The result means n³ and n leave the same remainder when divided by 6. It can simplify large cube remainder questions without calculating the full cube.
Build the explanation by naming the difference n cubed minus n and the factorisation n times n minus one times n plus one. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.
Repair the nearest misconception by checking three consecutive integers and returning to one factor 2 and one factor 3. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 8 OF 12 · Back to contents
8. Control examination working
Factor completely, label the three consecutive factors, identify where 2 and 3 come from, and conclude divisibility by 6. Do not merely cite several examples.
Build the explanation by naming the difference n cubed minus n and the factorisation n times n minus one times n plus one. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.
Repair the nearest misconception by checking three consecutive integers and returning to one factor 2 and one factor 3. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 9 OF 12 · Back to contents
9. Worked examples
Worked case 1: Evaluate 2³−2.
Controlled answer. 6. Eight minus two equals six.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Identify the three consecutive factors. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “6” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck the difference n cubed minus n, the factorisation n times n minus one times n plus one, three consecutive integers, one factor 2 and one factor 3. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 2: Evaluate 5³−5.
Controlled answer. 120. One hundred twenty-five minus five equals one hundred twenty.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Factor as 4×5×6. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “120” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck the difference n cubed minus n, the factorisation n times n minus one times n plus one, three consecutive integers, one factor 2 and one factor 3. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 3: Factor n³−n.
Controlled answer. n(n−1)(n+1). First take out n, then factor n²−1.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Reorder the factors consecutively. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “n(n−1)(n+1)” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck the difference n cubed minus n, the factorisation n times n minus one times n plus one, three consecutive integers, one factor 2 and one factor 3. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 4: Why is one of three consecutive integers even?
Controlled answer. Parity alternates between odd and even. Any block of three must contain an even member.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Mark it in 7, 8, 9. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “Parity alternates between odd and even” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck the difference n cubed minus n, the factorisation n times n minus one times n plus one, three consecutive integers, one factor 2 and one factor 3. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 5: Why is one of three consecutive integers divisible by 3?
Controlled answer. Multiples of 3 occur every third integer. A block of three consecutive integers contains one residue 0 modulo 3.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Mark it in 10, 11, 12. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “Multiples of 3 occur every third integer” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck the difference n cubed minus n, the factorisation n times n minus one times n plus one, three consecutive integers, one factor 2 and one factor 3. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 6: Why does having factors 2 and 3 prove divisibility by 6?
Controlled answer. Two and three are coprime and their product is six. Both factors occur in the same integer product.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Avoid double-counting a single factor. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “Two and three are coprime and their product is six” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck the difference n cubed minus n, the factorisation n times n minus one times n plus one, three consecutive integers, one factor 2 and one factor 3. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 7: Does the rule hold for n=0?
Controlled answer. Yes. Zero cubed minus zero is zero, and zero is a multiple of six.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Write 0=6×0. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “Yes” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck the difference n cubed minus n, the factorisation n times n minus one times n plus one, three consecutive integers, one factor 2 and one factor 3. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 8: Does the rule hold for n=1?
Controlled answer. Yes. One cubed minus one is zero.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Connect to the factorisation. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “Yes” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck the difference n cubed minus n, the factorisation n times n minus one times n plus one, three consecutive integers, one factor 2 and one factor 3. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 9: What remainder does 101³−101 leave when divided by 6?
Controlled answer. 0. The identity applies to every whole number.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Avoid expanding the cube. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “0” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck the difference n cubed minus n, the factorisation n times n minus one times n plus one, three consecutive integers, one factor 2 and one factor 3. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 10: What can be said about the remainders of n³ and n modulo 6?
Controlled answer. They are the same. Their difference is divisible by 6.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Test n=8. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “They are the same” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck the difference n cubed minus n, the factorisation n times n minus one times n plus one, three consecutive integers, one factor 2 and one factor 3. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 11: Repair: n³−n is divisible by 6 because cubes are multiples of 6.
Controlled answer. Cubes are not always multiples of 6; the factored difference contains three consecutive factors. For example, 5³ is not divisible by 6.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. State the correct mechanism. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “Cubes are not always multiples of 6; the factored difference contains three consecutive factors” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck the difference n cubed minus n, the factorisation n times n minus one times n plus one, three consecutive integers, one factor 2 and one factor 3. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 12: Repair: checking n=1 to 10 proves the rule for all n.
Controlled answer. Finite examples support a pattern but do not prove every case. The factorisation supplies the general proof.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Name the universal step. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “Finite examples support a pattern but do not prove every case” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck the difference n cubed minus n, the factorisation n times n minus one times n plus one, three consecutive integers, one factor 2 and one factor 3. Decide whether the evidence proves the conclusion or only supports a limited statement.
CHAPTER 10 OF 12 · Back to contents
10. Independent practice
Practice 1: Can the result be negative for a non-negative whole number n?
Explained answer. No. For n≥1, n³≥n, and at zero the difference is zero.
Delayed transfer. Keep the requested domain visible. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
Practice 2: Is n³+n always divisible by 6?
Explained answer. Not always. For n=1, the value is two.
Delayed transfer. Do not transfer the minus result blindly. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
Practice 3: What are the three factors when n=20?
Explained answer. 19, 20 and 21. They are n−1, n and n+1.
Delayed transfer. Identify factors 2 and 3. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
Practice 4: How can the identity check a cube calculation?
Explained answer. Subtract the base from the computed cube and confirm the result is divisible by 6. Failure signals an arithmetic error, though passing does not prove the cube is correct.
Delayed transfer. Audit 12³. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
Practice 5: What is the nearest wrong shortcut?
Explained answer. Claiming three consecutive numbers are all divisible by 6. Only their product is guaranteed to contain factors 2 and 3.
Delayed transfer. State the exact guarantee. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
Practice 6: How should the proof be ordered?
Explained answer. Factor n³−n, identify three consecutive factors, locate factors 2 and 3, then combine them. Each line has a distinct job.
Delayed transfer. Rebuild the proof from memory. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
Practice 7: Why is the factorisation stronger than examples?
Explained answer. It represents every whole number n at once. No untested case remains outside the algebraic form.
Delayed transfer. Explain the domain. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
Practice 8: State the full method.
Explained answer. Rewrite n³−n as (n−1)n(n+1), use the even member for a factor 2 and the multiple of 3 for a factor 3, and conclude the product is divisible by 6. Use the result to compare cube and base remainders without reversing unrelated claims.
Delayed transfer. Solve a large-number case. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
CHAPTER 11 OF 12 · Back to contents
11. Delayed transfer
On day one, solve clear examples. On day three, mix examples, near-nonexamples and incomplete evidence. On day seven, solve without headings or a word bank. Delay reveals whether the method can be reconstructed independently.
Keep an error log for the first failed stage: reading, representation, rule, execution or checking. The earliest failure identifies the repair.
Interleave this skill with neighbouring ideas that share vocabulary. Classify each item only after naming the distinguishing condition.
In timed work, mark the relationship, apply the rule, write the conclusion and audit one condition. If checking changes the answer, record the overlooked feature.
Create a teach-back card with a near-nonexample on the front and the repaired mechanism on the back. Explain why the tempting answer fails.
Finish with one basic example, one boundary case and one transfer case. Supply explanations that a classmate can verify.
For a second delayed test, reverse the direction: start from a conclusion and invent evidence that would justify it, then evidence that would be insufficient. This exposes the boundary between relevance and proof.
CHAPTER 12 OF 12 · Back to contents
12. Family guidance and FAQs
Guidance for parents and tutors
Ask for the decisive condition before confirming the answer. Use prompts such as “What exactly happened?”, “Which stage or meaning is this?” and “What would prove it?”
Fade support from labelled examples to mixed questions and delayed open response. Independent retrieval after time and topic change is the target.
Praise precise claim control. Narrowing an answer when evidence is incomplete is scientific and academic strength.
Frequently asked questions
Should I memorise the rule?
Memorise the compact rule, then practise examples and near-nonexamples so the conditions remain visible.
What if two answers seem possible?
Return to the exact relationship and choose the option satisfying every stated condition without adding facts.
Why delay practice?
Immediate repetition may depend on the recent explanation; delay tests reconstruction.
How should an error be corrected?
Name the first failed stage and practise it with a changed example.
When is the skill ready?
When the learner can explain, apply and transfer it after a delay.
What is the final audit?
Check relationship, mechanism, claim strength and whole-question fit.
Next routes
Continue at the Mathematics learning hub or return to the series directory.